No existence of the geometric potential for a Dirac fermion on two-dimensional curved surfaces of revolution

TL;DR

This paper proves that for Dirac fermions on two-dimensional curved surfaces of revolution, no curvature-induced quantum potential exists, contrasting with non-relativistic cases and aligning with experimental observations in topological insulators.

Contribution

The authors demonstrate within the Dirac quantization framework that relativistic particles on curved surfaces do not experience curvature-induced quantum potentials, a novel theoretical result.

Findings

No curvature-induced quantum potential for Dirac fermions on curved surfaces.

Contrasts with non-relativistic particles where such potentials exist.

Aligns with experimental evidence from topological insulators.

Abstract

For a free particle that non-relativistically moves on a curved surface, there are curvature-induced quantum potentials that significantly influence the surface quantum states, but the experimental results in topological insulators, whenever curved or not, indicate no evidence of such a potential, implying that there does not exist such a quantum potential for the relativistic particles, constrained on the surface or not. Within the framework of Dirac quantization scheme, we demonstrate a general result that for a Dirac fermion on a two-dimensional curved surface of revolution, no curvature-induced quantum potential is permissible.